Raising Exponential Order
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Theorem
Let $\map f t: \R \to \mathbb F$ a function, where $\mathbb F \in \set {\R, \C}$.
Let $f$ be continuous on the real interval $\hointr 0 \to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$.
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Let $f$ be of exponential order $a$.
Let $b > a$.
Then $f$ is of exponential order $b$.
Proof
From the definition of exponential order, there exist strictly positive real numbers $M$ and $K$ such that:
- $\forall t \ge M: \size {\map f t} < K e^{a t}$
From Exponential is Strictly Increasing, we have:
- $K e^{a t} < K e^{b t}$
Therefore:
- $\forall t \ge M: \size {\map f t} < K e^{b t}$
The result follows from the definition of exponential order.
$\blacksquare$